## Twistor transforms of quaternionic functions and orthogonal complex structures.(English)Zbl 1310.53045

The authors study orthogonal complex structures (OCS) on the space $$(\mathbb R^4=\mathbb H,\mathrm{can})$$ with the standard flat metric can. Let $$S=\{q\in\mathbb H:q^2=-1\}$$ be the sphere of imaginary unit quaternions. An orthogonal almost complex structure defined on an open subset $$\Omega\subset\mathbb R^4=\mathbb H$$ is simply a map $$J:\Omega\to S$$. The authors define a special OCS structure $$\mathbb J$$ on $$\Omega=\mathbb H-\mathbb R$$ as $$\mathbb J_q=\frac{\mathrm{Im}(q)}{|\mathrm{Im}(q)|}\in S$$. Let $$\Omega$$ be a domain in $$\mathbb H$$ and for $$I\in S$$ let us denote $$\Omega_I=\Omega\cap L_I$$ where $$L_I=\mathbb R+I\mathbb R$$. A function of the quaternion variable $$f:\Omega\to\mathbb H$$ is called regular if for all $$I\in S$$ the restriction $$f_{\Omega_I}$$ is holomorphic. Let $$\Omega$$ be a symmetric slice domain and let $$f:\Omega\to\mathbb H$$ be an injective regular function. Then $$f$$ induces an OCS structure $$\mathbb J^f=f_*\mathbb Jf_*^{-1}$$ on $$f(\Omega-\mathbb R)$$. The authors study OCS induced by regular functions $$f$$ of the quaternion variable. The twistor transform of $$f$$ corresponds to a holomorphic curve in a Klein quadric. “The case in which $$\Omega$$ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space $$\mathbb {CP}^3$$.”

### MSC:

 53C28 Twistor methods in differential geometry 30G35 Functions of hypercomplex variables and generalized variables 53C55 Global differential geometry of Hermitian and Kählerian manifolds 14J26 Rational and ruled surfaces
Full Text:

### References:

 [1] Bishop, E.: Conditions for the analyticity of certain sets. Michigan Math. J. 11, 289-304 (1964) · Zbl 0143.30302 · doi:10.1307/mmj/1028999180 [2] Bisi, C., Gentili, G.: Möbius transformations and the Poincaré distance in the quaternionic setting. Indiana Univ. Math. J. 58, 2729-2764 (2009) · Zbl 1193.30067 · doi:10.1512/iumj.2009.58.3706 [3] Chen, J., Li, J.: Quaternionic maps between hyperkähler manifolds. J. Differential Geom. 55, 355-384 (2000) · Zbl 1067.53035 [4] Colombo, F., Gentili, G., Sabadini, I., Struppa, D.: Extension results for slice regular func- tions of a quaternionic variable. Adv. Math. 222, 1793-1808 (2009) · Zbl 1179.30052 · doi:10.1016/j.aim.2009.06.015 [5] Colombo, F., Sabadini, I., Sommen, F., Struppa, D. C.: Analysis of Dirac Systems and Computational Algebra. Progr. Math. Phys. 39, Birkhäuser Boston, Boston, MA (2004) · Zbl 1064.30049 [6] Conway, J. B.: Functions of One Complex Variable. 2nd ed., Grad. Texts in Math. 11, Springer, New York (1978) · Zbl 0277.30001 [7] Cullen, C. G.: An integral theorem for analytic intrinsic functions on quaternions. Duke Math. J. 32, 139-148 (1965) · Zbl 0173.09001 · doi:10.1215/S0012-7094-65-03212-6 [8] de Bartolomeis, P., Nannicini, A.: Introduction to the differential geometry of twistor spaces. In: Geometric Theory of Singular Phenomena in P.D.E., Symposia Math. 38, Cambridge Univ. Press, 91-160 (1998) · Zbl 0921.53035 [9] Dolgachev, I. V.: Classical Algebraic Geometry. A Modern View. Cambridge Univ. Press, Cambridge (2012) · Zbl 1252.14001 · doi:10.1017/CBO9781139084437 [10] Edge, W.: The Theory of Ruled Surfaces. Cambridge Univ. Press (1931) · Zbl 0627.58019 [11] Gentili, G., Stoppato, C.: Zeros of regular functions and polynomials of a quaternionic vari- able. Michigan Math. J. 56, 655-667 (2008) · Zbl 1184.30048 · doi:10.1307/mmj/1231770366 [12] Gentili, G., Stoppato, C.: The open mapping theorem for regular quaternionic functions. Ann. Scuola Norm. Sup. Pisa 8, 805-815 (2009) · Zbl 1201.30067 · doi:10.2422/2036-2145.2009.4.07 [13] Gentili, G., Stoppato, C.: The zero sets of slice regular functions and the open mapping theo- rem. In: Hypercomplex Analysis and Applications, I. Sabadini and F. Sommen (eds.), Trends in Math., Birkhäuser, Basel, 95-107 (2011) · Zbl 1234.30039 · doi:10.1007/978-3-0346-0246-4_7 [14] Gentili, G., Struppa, D. C.: A new theory of regular functions of a quaternionic variable. Adv. Math. 216, 279-301 (2007) · Zbl 1124.30015 · doi:10.1016/j.aim.2007.05.010 [15] Gentili, G., Struppa, D. C.: On the multiplicity of zeroes of polynomials with quaternionic coefficients. Milan J. Math. 76, 15-25 (2008) · Zbl 1194.30054 · doi:10.1007/s00032-008-0093-0 [16] Gentili, G., Struppa, D. C., Vlacci, F.: The fundamental theorem of algebra for Hamilton and Cayley numbers. Math. Z. 259, 895-902 (2008) · Zbl 1144.30004 · doi:10.1007/s00209-007-0254-9 [17] Joyce, D.: Hypercomplex algebraic geometry. Quart. J. Math. Oxford 49, 129-162 (1998) · Zbl 0924.14002 · doi:10.1093/qjmath/49.194.129 [18] Mumford, D.: Algebraic Geometry. I. Classics in Math., Springer, Berlin (1995) · Zbl 0821.14001 [19] Polo-Blanco, I., van der Put, M., Top, J.: Ruled quartic surfaces, models and classification. Geom. Dedicata 150, 151-180 (2011) · Zbl 1226.14048 · doi:10.1007/s10711-010-9500-0 [20] Quillen, D.: Quaternionic algebra and sheaves on the Riemann sphere. Quart. J. Math. Oxford 49, 163-198 (1998) · Zbl 0921.14003 · doi:10.1093/qjmath/49.194.163 [21] Salamon, S.: Harmonic and holomorphic maps. In: Geometry Seminar Luigi Bianchi II - 1984, Lecture Notes in Math. 1164, Springer, Berlin, 161-224 (1985) · Zbl 0591.53031 [22] Salamon, S. M.: Quaternionic manifolds. In: Symposia Matematica, Vol. 26 (Rome, 1980), Academic Press, London, 139-151 (1982) · Zbl 0534.53030 [23] Salamon, S., Viaclovsky, J.: Orthogonal complex structures on domains in 4 R . Math. Ann. 343, 853-899 (2009); see also · Zbl 1167.32017 · doi:10.1007/s00208-008-0293-5 [24] Shapiro, G.: On discrete differential geometry in twistor space. J. Geom. Phys. 68, 81-102 (2013) · Zbl 1283.53013 · doi:10.1016/j.geomphys.2013.02.008 [25] Stoppato, C.: Regular Moebius transformations of the space of quaternions. Ann. Global Anal. Geom. 39, 387-401 (2011) · Zbl 1214.30044 · doi:10.1007/s10455-010-9238-9 [26] Stoppato, C.: A new series expansion for slice regular functions. Adv. Math. 231, 1401-1416 (2012) · Zbl 1262.30059 · doi:10.1016/j.aim.2012.05.023 [27] Stoppato, C.: Singularities of slice regular functions. Math. Nachr. 285, 1274-1293 (2012) · Zbl 1253.30076 · doi:10.1002/mana.201100082 [28] Sudbery, A.: Quaternionic analysis. Math. Proc. Cambridge Philos. Soc. 85, 199-224 (1979) · Zbl 0399.30038 · doi:10.1017/S0305004100055638 [29] Wood, J. C.: Harmonic morphisms and Hermitian structures on Einstein 4-manifolds. Int. J. Math. 3, 415-439 (1992) · Zbl 0763.53051 · doi:10.1142/S0129167X92000187
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.